Objective. Evaluate the effectiveness of cognitive-behavior therapy for chronic fatigue syndrome.

Participant pool. 142 patients who were recruited from referrals by primary care physicians and consultants to a hospital clinic specializing in chronic fatigue syndrome.

Actual participants. Only 60 of the 142 referred patients entered the study. Some were excluded because they didn't meet the diagnostic criteria, some had other health issues, and some refused to be a part of the study.

Patients randomly assigned to treatment and control groups, 30 patients in each group:

Treatment: Cognitive behavior therapy \(-\) collaborative, educative, and with a behavioral emphasis. Patients were shown on how activity could be increased steadily and safely without exacerbating symptoms.

Control: Relaxation \(-\) No advice was given about how activity could be increased. Instead progressive muscle relaxation, visualization, and rapid relaxation skills were taught.

The table below shows the distribution of patients with good outcomes at 6-month follow-up. Note that 7 patients dropped out of the study: 3 from the treatment and 4 from the control group.

Treatment Group: 19/27 = 0.70 = 70%

Control Group: 5/26 = 0.19 = 19%

Do the data show a "real" difference between the groups?

Suppose you flip a coin 100 times. While the chance a coin lands heads in any given coin flip is 50%, we probably won't observe exactly 50 heads. This type of fluctuation is part of almost any type of data generating process.

The observed difference between the two groups (70 - 19 = 51%) may be real, or may be due to natural variation.

Since the difference is quite large, it is more believable that the difference is real.

We use statistical tools to determine if the difference is so large that we should reject the notion that it was due to chance.

Are the results of this study generalizable to all patients with chronic fatigue syndrome?

No. These patients had specific characteristics and volunteered to be a part of this study, therefore they may not be representative of all patients with chronic fatigue syndrome.

While we cannot immediately generalize the results to all patients, this first study is encouraging. The method at least works for patients with some narrow set of characteristics, and that gives hope that it will work, at least to some degree, with other patients.

Data collected on students in a statistics class on a variety of variables

• gender \(-\) categorical
• sleep \(-\) numerical, continuous
• bedtime \(-\) categorical, ordinal
• countries \(-\) numerical, discrete
• dread \(-\) categorical, ordinal (could also be used as numerical)

What type of variable is a telephone area code?

1. numerical, continuous
2. numerical, discrete
3. categorical
4. categorical, ordinal

What type of variable is a telephone area code?

1. numerical, continuous
2. numerical, discrete
3. categorical
4. categorical, ordinal

Does there appear to be a relationship between the hours of study per week and the GPA of a student?

Can you spot anything unusual about any of the data points?

Based on the scatterplot, which of the following statements is correct about the head and skull length of possums?

1. There is no relationship between head length and skull width, i.e. the variables are independent.
2. Head length and skull width are positively associated.
3. Skull width and head length are negatively associated.
4. A longer head causes the skull to be wider.
5. A wider skull causes the head to be longer.

Based on the scatterplot, which of the following statements is correct about the head and skull length of possums?

1. There is no relationship between head length and skull width, i.e. the variables are independent.
2. Head length and skull width are positively associated.
3. Skull width and head length are negatively associated.
4. A longer head causes the skull to be wider.
5. A wider skull causes the head to be longer.

• When two variables show some connection with one another, they are called associated variables.
• Associated variables can also be called dependent variables, and vice-versa
• If two variables are not associated, i.e., there is no evident connection between the two, then they are said to be independent.

Research Question Can people become better, more efficient runners on their own, merely by running?

Population of Interest All people

Sample Group of adult women who recently joined a running group

Population to which results can be generalized Adult women, if the data are randomly sampled

• Anti-smoking research started in the 1930s and 1940s when cigarette smoking became increasingly popular. While some smokers seemed to be sensitive to cigarette smoke, others were completely unaffected.
• Anti-smoking research was faced with resistance based on anecdotal evidence such as "My uncle smokes three packs a day and he's in perfectly good health", evidence based on a limited sample size that might not be representative of the population.
• It was concluded that "smoking is a complex human behavior, by its nature difficult to study, confounded by human variability."
• In time researchers were able to examine larger samples of cases (smokers), and trends showing that smoking has negative health impacts became much clearer.

• Wouldn't it be better to just include everyone and "sample" the entire population?
• This is called a census
• There are problems with taking a census:
• It can be difficult to complete a census: there always seem to be some individuals who are hard to locate or hard to measure. And these difficult-to-find people may have certain characteristics that distinguish them from the rest of the population.
• Populations rarely stand still. Even if you could take a census, the population changes constantly, so it's never possible to get a perfect measure.
• Taking a census may be more complex than sampling.

• Sampling is natural.
• Think about sampling something you are cooking - you taste (examine) a small part of what you're cooking to get an idea about the dish as a whole.
• When you taste a spoonful of soup and decide the spoonful you tasted isn't salty enough, that's exploratory analysis.
• If you generalize and conclude that your entire soup needs salt, that's an inference.
• For your inference to be valid, the spoonful you tasted (the sample) needs to be representative of the entire pot (the population).
• If your spoonful comes only from the surface and the salt is collected at the bottom of the pot, what you tasted is probably not representative of the whole pot.
• If you first stir the soup thoroughly before you taste, your spoonful will more likely be representative of the whole pot.

Non-response: If only a small fraction of the randomly sampled people choose to respond to a survey, the sample may no longer be representative of the population.

Voluntary response: Occurs when the sample consists of people who volunteer to respond because they have strong opinions on the issue. Such a sample will also not be representative of the population.

Convenience sample: Individuals who are easily accessible are more likely to be included in the sample.

A historical example of a biased sample yielding misleading results. In 1936, Alf Landon became the Republican presidential nominee, opposing the re-election of Franklin Delano Roosevelt, a Democrat.

Landon (GOP)

FDR (DEM)

• The Literary Digest polled about 10 million Americans, and got responses from about 2.4 million.
• The poll showed that Landon would likely be the overwhelming winner and FDR would get only 43% of the votes.
• Election result: FDR won, with 62% of the votes (and 98.5% of the electoral votes \(-\) the most lop-sided electoral vote victory in US history).
• The magazine was completely discredited because of the poll, and was soon discontinued.

• The magazine had surveyed
• its own readers,
• registered automobile owners, and
• registered telephone users.

These groups had incomes well above the national average of the day (remember, this is Great Depression era) which resulted in lists of voters far more likely to support Republicans than a truly typical voter of the time, i.e., the sample was not representative of the American population at the time.

• The Literary Digest election poll was based on a sample size of 2.4 million, which is huge, but since the sample was biased, the sample did not yield an accurate prediction.
• Back to the soup analogy: If the soup is not well stirred, it doesn't matter how large a spoon you have, it will still not taste right. If the soup is well stirred, a small spoon will suffice to test the soup.

• To identify the explanatory variable in a pair of variables, identify which of the two is suspected of affecting the other:
  \[ \text{explanatory variable} \overset{\text{might affect}}{\longrightarrow} \text{response variable} \]
• Labeling variables as explanatory and response does not guarantee the relationship between the two is actually causal, even if there is an association identified between the two variables. We use these labels only to keep track of which variable we suspect affects the other.

What type of study is this: observational or experiment?

"Girls who regularly ate breakfast, particularly one that includes cereal, were slimmer than those who skipped the morning meal, according to a study that tracked nearly 2,400 girls for 10 years. […] As part of the survey, the girls were asked once a year what they had eaten during the previous three days."

This is an observational study since the researchers merely observed the behavior of the girls (subjects) as opposed to imposing treatments on them.

What is the conclusion of the study?

There is an association between girls eating breakfast and being slimmer.

Who sponsored the study?

General Mills (cereal manufacturer)

1. Eating breakfast causes girls to be thinner.
   

1. Being thin causes girls to eat breakfast.

   

2. A third variable is responsible for both. What could it be? An extraneous variable that affects both the explanatory and the response variable and that make it seem like there is a relationship between the two is called a confounding variable.

   

A prospective study identifies individuals and collects information as events unfold.

• Example: The Nurses Health Study has been recruiting registered nurses and then collecting data from them using questionnaires since 1976.

Retrospective studies collect data after events have taken place.

• Example: Researchers reviewing past events in medical records.

Almost all statistical methods are based on the notion of implied randomness.

If observational data are not collected in a random framework from a population, these statistical methods – the estimates and errors associated with the estimates – are not reliable.

Most commonly used random sampling techniques are simple, stratified, and cluster sampling.

Randomly select cases from the population, where there is no implied connection between the points that are selected.

Strata are made up of similar observations. We take a simple random sample from each stratum.

Clusters are usually not made up of homogeneous observations, and we take a simple random sample from a random sample of clusters. Usually preferred for economical reasons.

A city council has requested a household survey be conducted in a suburban area of their city. The area is broken into many distinct and unique neighborhoods, some including large homes, some with only apartments. Which approach would likely be the least effective?

1. Simple random sampling
2. Cluster sampling
3. Stratified sampling
4. Blocked sampling

A city council has requested a household survey be conducted in a suburban area of their city. The area is broken into many distinct and unique neighborhoods, some including large homes, some with only apartments. Which approach would likely be the least effective?

1. Simple random sampling
2. Cluster sampling
3. Stratified sampling
4. Blocked sampling

Studies where researchers assign treatments to cases are called experiments. If the assignment of treatments to cases (e.g., using a coin flip to determine which treatment a patient receives), the study is called a randomized experiment.

Randomized experiments have a series of four principles.

Researchers assign treatments to cases, and do their best to control for other differences between groups.

Example: in a drug trial, patients may be asked to take a pill daily. Some may take the pill dry (ick!), some with just a sip of water, some with coffee, and others with juice. To control for the effect of accompanying liquid, a doctor may ask all patients to drink a 12 oz glass of water with the pill.

Researchers randomize patients into treatment groups to account for variables that cannot be controlled.

Example: some patients are more susceptible to disease than others due to dietary habits. Randomizing patients into treatment/control groups helps even out these differences, possibly preventing accidental bias.

The more cases researchers observe, the more accurately they can estimate the effects of explanatory variables on response variables. In a single study, we replicate by collecting a sufficiently large sample. Additionally, scientists often replicate an entire study over again to verify earlier findings.

Researchers sometimes know (or suspect) that variables other than the treatment influence the response. Under this situation, they may first group individuals by this variable, and then randomize cases within each block. This is known as blocking.

Example: If we were researching the effect of a drug on heart attacks, we might first split patients into high-risk and low-risk blocks (based on diet, physique, genetic screening, or some other approach), and then randomly assign half of each block to the control group, and the other half to the drug (treatment) group.

A study is designed to test the effect of light level and noise level on exam performance of students. The researcher also believes that light and noise levels might have different effects on males and females, so wants to make sure both genders are equally represented in each group. Which of the below is correct?

1. There are 3 explanatory variables (light, noise, gender) and 1 response variable (exam performance)
2. There are 2 explanatory variables (light and noise), 1 blocking variable (gender), and 1 response variable (exam performance)
3. There is 1 explanatory variable (gender) and 3 response variables (light, noise, exam performance)
4. There are 2 blocking variables (light and noise), 1 explanatory variable (gender), and 1 response variable (exam performance)

A study is designed to test the effect of light level and noise level on exam performance of students. The researcher also believes that light and noise levels might have different effects on males and females, so wants to make sure both genders are equally represented in each group. Which of the below is correct?}

1. There are 3 explanatory variables (light, noise, gender) and 1 response variable (exam performance)
2. There are 2 explanatory variables (light and noise), 1 blocking variable (gender), and 1 response variable (exam performance)
3. There is 1 explanatory variable (gender) and 3 response variables (light, noise, exam performance)
4. There are 2 blocking variables (light and noise), 1 explanatory variable (gender), and 1 response variable (exam performance)

Factors are conditions we can impose on the experimental units.

Blocking variables are characteristics that the experimental units come with, that we would like to control for.

Blocking is like stratifying, except used in experimental settings when randomly assigning, as opposed to when sampling.

Placebo: fake treatment, often used as the control group for medical studies

Placebo effect: experimental units showing improvement simply because they believe they are receiving a special treatment

Blinding: when experimental units do not know whether they are in the control or treatment group

Double-blind: when both the experimental units and the researchers who interact with the patients do not know who is in the control and who is in the treatment group

What is the main difference between observational studies and experiments?

1. Experiments take place in a lab while observational studies do not need to.
2. In an observational study we only look at what happened in the past.
3. Most experiments use random assignment while observational studies do not.
4. Observational studies are completely useless since no causal inference can be made based on their findings.

What is the main difference between observational studies and experiments?

1. Experiments take place in a lab while observational studies do not need to.
2. In an observational study we only look at what happened in the past.
3. Most experiments use random assignment while observational studies do not.
4. Observational studies are completely useless since no causal inference can be made based on their findings.

Scatterplots are useful for visualizing the relationship between two numerical variables.

Do life expectancy and total fertility appear to be associated or independent?

They appear to be linearly and negatively associated: as fertility increases, life expectancy decreases.

Was the relationship the same throughout the years, or did it change?

The relationship changed over the years.

Useful for visualizing one numerical variable. Darker colors represent areas where there are more observations.

How would you describe the distribution of GPAs in this data set?

Make sure to say something about the center, shape, and spread of the distribution.

The mean, also called the average (marked with a triangle in the plot), is one way to measure the center of a distribution of data.

The mean GPA is 3.59.

The sample mean, denoted as \(\bar{x}\), can be calculated as \[ \bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n} = \frac{1}{n} \sum_{i=1}^{n} x_i \] where \(x_1, x_2, \cdots, x_n\) represent the \(n\) observed values.

The population mean is also computed the same way but is denoted as \(\mu\). It is often not possible to calculate \(\mu\) since population data are rarely available.

The sample mean is a sample statistic, and serves as a point estimate of the population mean. This estimate may not be perfect, but if the sample is good (representative of the population), it is usually a pretty good estimate.

Higher bars represent areas where there are more observations, makes it a little easier to judge the center and the shape of the distribution.

• Histograms provide a view of the data density. Higher bars represent where the data are relatively more common.
• Histograms are especially convenient for describing the shape of the data distribution.
• The chosen bin width can alter the story the histogram is telling.

Which one(s) of these histograms are useful? Which reveal too much about the data? Which hide too much?

Does the histogram have a single prominent peak (unimodal), several prominent peaks (bimodal/multimodal), or no apparent peaks (uniform)?

Note: In order to determine modality, step back and imagine a smooth curve over the histogram – imagine that the bars are wooden blocks and you drop a limp spaghetti over them, the shape the spaghetti would take could be viewed as a smooth curve.

Is the histogram right skewed, left skewed or symmetric?

Histograms are said to be skewed to the side of the long tail.

Are there any unusual observations or potential outliers?

Which of these variables do you expect to be uniformly distributed?

1. weights of adult females
2. salaries of a random sample of people from North Carolina
3. house prices
4. birthdays of classmates (day of the month)

Which of these variables do you expect to be uniformly distributed?

1. weights of adult females
2. salaries of a random sample of people from North Carolina
3. house prices
4. birthdays of classmates (day of the month)

Sketch the expected distributions of the following variables:

• number of piercings
• scores on an exam
• IQ scores

Come up with a concise way (1-2 sentences) to teach someone how to determine the expected distribution of any variable.

How useful are centers alone for conveying the true characteristics of a distribution?

Variance is roughly the average squared deviation from the mean.

\[ s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]

• The sample mean is \(\bar{x} = 6.71\), and the sample size is \(n = 217\).
• The variance of the amount of sleep students get per night can be calculated as:

\[ \begin{split} s^2 &= \frac{1}{217 - 1} \left[ (5-6.71)^2 + (9-6.71)^2 + \cdots + (7-6.71)^2 \right] \\ &= 4.11 \text{ hours}^2 \end{split} \]

Why do we use the squared deviation in the calculation of variance?

• To get rid of negatives so that observations equally distant from the mean are weighed equally.
• To weigh larger deviations more heavily.

The standard deviation is the square root of the variance, and has the same units as the data.

\[ s = \sqrt{s^2} \]

The standard deviation of the amount of sleep students get per night can be calculated as: \[ s = \sqrt{4.11} = 2.03 \text{ hours} \] We can see that all of the data are within 3 standard deviations of the mean of \(\bar{x} = 6.17\).

The median is the value that splits the data in half when ordered in ascending order.

If there are an even number of observations, then the median is the average of the two values in the middle.

Since the median is the midpoint of the data, 50% of the values are below it. Hence, it is also the 50th percentile.

A percentile is the the smallest value from an ordered list of numbers which is greater than or equal to that percentage of list elements.

Example: The \(42^\text{nd}\) percentile of the numbers \(\{ 1, 2, 3, \cdots, 99, 100 \}\) is 42.

It can become quite complicated when there aren't an even multiple of 100 items!

• The 25th percentile is also called the first quartile, Q1.
• The 50th percentile is also called the median.
• The 75th percentile is also called the third quartile, Q3.

Between Q1 and Q3 is the middle 50% of the data. The range these data span is called the interquartile range, or the IQR.

\[ \text{IQR} = \text{Q3} - \text{Q1} \]

The box in a box plot represents the middle 50% of the data, and the thick line in the box is the median.

The whiskers of a box plot can extend up to \(1.5 \times \text{IQR}\) away from the quartiles.

• max upper whisker reach = \(\text{Q3} + 1.5 \times \text{IQR}\)
• max lower whisker reach = \(\text{Q1} - 1.5 \times \text{IQR}\)

Example: IQR: 20 - 10 = 10

• max upper whisker reach = \(20 + 1.5 \times 10 = 35\)
• max lower whisker reach = \(10 - 1.5 \times 10 = -5\)

A potential outlier is defined as an observation beyond the maximum reach of the whiskers. It is an observation that appears extreme relative to the rest of the data.

Why is it important to look for outliers?

• Identify extreme skew in the distribution.
• Identify data collection and entry errors.
• Provide insight into interesting features of the data.

How would sample statistics such as mean, median, SD, and IQR of household income be affected if the largest value was replaced with $10 million? What if the smallest value was replaced with $10 million?

Median and IQR are more robust to skewness and outliers than mean and SD. Therefore,

• for skewed distributions it is often more helpful to use median and IQR to describe the center and spread
• for symmetric distributions it is often more helpful to use the mean and SD to describe the center and spread

If you would like to estimate the typical household income for a student, would you be more interested in the mean or median income?

Median and IQR are more robust to skewness and outliers than mean and SD. Therefore,

• for skewed distributions it is often more helpful to use median and IQR to describe the center and spread
• for symmetric distributions it is often more helpful to use the mean and SD to describe the center and spread

If you would like to estimate the typical household income for a student, would you be more interested in the mean or median income?

Median

If the distribution is symmetric, center is often defined as the mean:

• mean \(\approx\) median

If the distribution is skewed or has extreme outliers, center is often defined as the median

• Right-skewed: mean > median
• Left-skewed: mean < median

Which is most likely true for the distribution of percentage of time actually spent taking notes in class versus on Facebook, Twitter, etc.?

1. mean > median    3. mean \(\approx\) median
2. mean < median    4. impossible to tell

Which is most likely true for the distribution of percentage of time actually spent taking notes in class versus on Facebook, Twitter, etc.?

1. mean > median    3. mean \(\approx\) median
2. mean < median    4. impossible to tell

When data are extremely skewed, transforming them might make modeling easier. A common transformation is the log transformation.

The histograms on the left shows the distribution of number of basketball games attended by students. The histogram on the right shows the distribution of log of number of games attended.

Skewed data are easier to model with when they are transformed because outliers tend to become far less prominent after an appropriate transformation.

However, results of an analysis might be difficult to interpret because the log of a measured variable is usually meaningless.

What other variables would you expect to be extremely skewed?

Skewed data are easier to model with when they are transformed because outliers tend to become far less prominent after an appropriate transformation.

However, results of an analysis might be difficult to interpret because the log of a measured variable is usually meaningless.

What other variables would you expect to be extremely skewed?

Salary, housing prices, ability to throw a football, …

What patterns are apparent in the change in population between 2000 and 2010?

A table that summarizes data for two categorical variables is called a contingency table.

The contingency table below shows the distribution of students' genders and whether or not they are looking for a spouse while in college.

A bar plot is a common way to display a single categorical variable. A bar plot where proportions instead of frequencies are shown is called a relative frequency bar plot.

How are bar plots different than histograms? Bar plots are used for displaying distributions of categorical variables, while histograms are used for numerical variables. The x-axis in a histogram is a number line, hence the order of the bars cannot be changed, while in a bar plot the categories can be listed in any order (though some orderings make more sense than others, especially for ordinal variables.)

Does there appear to be a relationship between gender and whether the student is looking for a spouse in college?

To answer this question we examine the row proportions:

• % Females looking for a spouse: 51 / 137 ~ 0.37
• % Males looking for a spouse: 18 / 70 ~ 0.26

What are the differences between the three visualizations shown below?

Can you tell which order encompasses the lowest percentage of mammal species?

http://www.bucknell.edu/msw3

Does there appear to be a relationship between class year and the number of clubs students are in?